Stable Iterative Methods for the Inversion of Geophysical Data

Abstract

Interpretation of earth electrical measurements can often be assisted by inversion, which is a non-linear model-fitting problem in these cases. Iterative methods are normally used, and the solution is defined by ‘best fit’ in the sense of generalized least-squares. The inverse problems we describe are ill-posed. That is, small changes in the data can lead to large changes in both the solution and in the iterative process that finds the solution. Through an analysis of the problem, based on local linearization, we define a class of methods that stabilize the iteration, and provide a robust solution. These methods are seen as generalizations of the well-known Singular Value Truncation and Marquardt Methods of iterative inversion. Here, and in a companion paper, we give examples illustrating the successful application of the method to ill-posed problems relating to the resistivity of the Earth.